Alternative steady states in ecological networks

In many natural situations, one observes a local system with many competing species that is coupled by weak immigration to a regional species pool. The dynamics of such a system is dominated by its stable and uninvadable (SU) states. When the competition matrix is random, the number of SUs depends on the average value and variance of its entries. Here we consider the problem in the limit of weak competition and large variance. Using a yes-no interaction model, we show that the number of SUs corresponds to the number of maximum cliques in an Erdös-Rényi network. The number of SUs grows exponentially with the number of species in this limit, unless the network is completely asymmetric. In the asymmetric limit, the number of SUs is $O\left(1\right)$. Numerical simulations suggest that these results are valid for models with a continuous distribution of competition terms.

Figure 1

Solutions for Eq. (14). The full, black line is the exact numerical solution, the dashed blue is $1-\alpha$, and the dotted red line depicts $ln\left(\alpha \right)/\alpha$.

Figure 2

$F({\beta }_0\left(\alpha \right))$, or $ln\left({\mathcal{N}}_{\text{SU}}\right)/N$, vs $\alpha$ for $0<\alpha <40$. The number of SUs grows exponentially with $N$, where the coefficient of the exponent is between 0 and 0.25.

Figure 3

$ln\left({\mathcal{N}}_{\text{SU}}\right)$ vs $ln\left(p\right)$, as obtained from a numerical summation of the Bollobás-Erdös formula for the symmetric case, for $N=20$ (red), $N=40$ (green), and $N=80$ (blue). The growth of the number of SUs with $N$ becomes exponential when $1-p$ is $O(1/N)$, as expected. When $1-p$ is $O(1/N^2)$, there is a drop toward one SU with all the $N$ species. In this regime, the number of SUs is independent of $N$.

Figure 4

Number of SUs (averaged over 500 samples) vs ${\sigma }_{\gamma }$ as obtained from an exact enumeration of all the stable and uninvadable combinations of species in the $\mathrm{\Gamma }$ model for $N=20$. Red points are actual results, each obtained from examination for stability and uninvadability of all the $2^{20}$ combinations of species; the dashed line is just to guide the eye. The $\mathrm{\Gamma }$ model is described by Eq. (3) with $C=1$ and where each pair of numbers $c_{i,j}=c_{j,i}$ is picked independently from a $\mathrm{\Gamma }$ distribution with mean 1 and standard deviation ${\sigma }_{\gamma }$. While the number of SUs is smaller than their number in the corresponding binomial model (we have suggested in [16] that the binomial model yields an upper bound for the $\mathrm{\Gamma }$), we still observe the growth in the number of SUs when ${\sigma }_{\gamma }\sim \sqrt{N}$ (around four to five), then it drops toward the full coexistence phase.

Figure 5

$ln\left({\mathcal{N}}_{\text{SU}}\right)$ vs $p$, as obtained from a numerical summation of the Bollobás-Erdös formula for the asymmetric case, for $N=1000$. In general, the number of SUs decays with $p$, with no exponential peak close to the fully connected limit. This general trend is superimposed on oscillations in the region where $p$ is $O\left(1\right)$, since in this regime there is only one integer value of maximal clique sizes that dominate the sum, as explained in the text. To demonstrate that, thick points were added to mark the $p$ values for which $S={\beta }_{0}N$ is 2, 3, 4, 5, and 6.

Figure 6

Number of SUs (averaged over 500 samples) vs ${\sigma }_{\gamma }$ as obtained from an exact enumeration of all the stable and uninvadable combinations of species in the asymmetric $\mathrm{\Gamma }$ model for $N=20$. Red points are actual results, each obtained from examination for stability and uninvadability of all the $2^{20}$ combinations of species; the dashed black line is just to guide the eye. The asymmetric $\mathrm{\Gamma }$ model is described by Eq. (3) with $C=1$ and where each $c_{i,j}$ is picked independently from a $\mathrm{\Gamma }$ distribution with mean 1 and standard deviation ${\sigma }_{\gamma }$.

Figure 7

The scaled most common clique size, ${\beta }_0$, as a function of $\alpha$ (the scaled value of $p$) for $\eta =0.5$.

Figure 8

The natural logarithm of the number of SUs as a function of $\eta$, for various $\alpha$, with $N=1000$. The linear approximation (40) (dashed lines) is valid close to $\eta =1$ for large $\alpha$'s, but over almost all the region for small $\alpha$'s.

Figure 9

The most common clique size, $S_0$, as a function of $p$ for $\eta =0.5$. Exact results are compared with the approximation (A7) for $N=100$ and 1000.